First and foremost, double check that you have the packages containing libssl.so, libcrypto.so and libmysqlclient_r.so installed (looks like you have this last one, it's extremely likely you also have the first two, but just double check).

Then, your problem is that you have those shared objects with a different SONAME, sign that they're binary incompatible with the plugin shipped with Qt, which therefore needs to be recompiled.

Therefore:

install the development versions of the packages found above (libssl-dev, mysql-client-dev or similar).

run the MaintenanceTool from your Qt installation, and be sure to select to install Qt's source code too.

Go in QTDIR/5.6/Src/qtbase/src/plugins/sqldrivers/mysql/.

Run the rightqmake, i.e. the one coming from that installation of Qt (not the system wide one or similar). Best way to be sure is providing the full path to it: QTDIR/5.6/gcc_64/bin/qmake.

Run make. Hopefully this will just work™; if it complains about some missing libraries, install them and rerun make.

This should now have produced a new libqsqlmysql.so plugin; overwrite the old one with this new one.